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The centre conditions for a certain cubic system. (English) Zbl 0899.34021
The authors prove the following theorem:
The system $\dot z= iz+ Az^2+ Dz^3+ Ez^2\overline z+ Fz\overline z^2\tag{1}$ (with $$z= x+ iy\in\mathbb{C}\simeq\mathbb{R}^2$$; $$A,D,E,F\in\mathbb{C}$$) has a centre at the origin iff one of the following three conditions holds:
1) $$E= F=0$$;
2) $$\text{Re}(E)= \text{Im}(DF)= \text{Re}(A^2\overline D)= \text{Re}(A^2F)= 0$$;
3) $$\text{Re}(E)= D- F=0$$.
The proof relies on
Lemma 1. If one of the conditions 1), 2), 3) holds then (1) has the centre $$z= 0$$.
Lemma 2. The first Lyapunov focus numbers of (1) are:
$$g_{11}= 2\text{ Re}(E)$$, $$g_{22}= \text{Im}(DF)\text{ mod}(g_{11})$$,
$$g_{33}= -\text{Im}(E)\cdot \text{Re}(A^2(\overline D- F))\text{ mod}(g_{11}, g_{22})$$,
$$g_{44}= -\text{Re}(A^2\overline F(\overline D- F)^2)\text{ mod}(g_{11}, g_{22}, g_{33})$$.
The proof of Lemma 2 is based on a method developed by V. G. Romanovskij [Differ. Equations 29, No. 5, 782-784 (1993); translation from Differ. Uravn. 29, No. 5, 910-912 (1993; Zbl 0833.34028)].

##### MSC:
 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations